ScalingStacks

Example 2.2. (1) The prototypical example of a weight structure arises from the stupid filtration of complexes. Let $\mathcal{A}$ be an idempotent-closed additive category. Then there is a bounded weight structure on $\operatorname{K}^{b}(\mathcal{A})$ given by

	$\displaystyle\operatorname{K}^{b}(\mathcal{A})^{w\geq 0}$	$\displaystyle=\langle X\mid X_{i}=0\text{ for all }i<0\rangle_{\cong}\,\,\,\,\text{ and}$
	$\displaystyle\operatorname{K}^{b}(\mathcal{A})^{w\leq 0}$	$\displaystyle=\langle X\mid X_{i}=0\text{ for all }i>0\rangle_{\cong},$

the subcategories generated by the complexes in non-negative and non-positive degrees under isomorphism. Most of the axioms of a weight structure are straigtforward to check. The idempotent-completeness $\operatorname{K}^{b}(\mathcal{A})^{w\geq 0}$ and $\operatorname{K}^{b}(\mathcal{A})^{w\leq 0}$ is discussed in [Sch11a]. The heart of the weight structure is $\operatorname{K}^{b}(\mathcal{A})^{w=0}=\mathcal{A}.$ In general, the heart of a weight structure is additive and idempotent closed but not necessarily abelian.
(2) Assume that $\mathcal{A}$ is abelian and that every object in $\mathcal{A}$ has a finite projective resolution. Then one can identify the bounded derived category of $\mathcal{A}$ with the bounded homotopy category of projectives

$$\operatorname{D}^{b}(\mathcal{A})=\operatorname{K}^{b}(\operatorname{Proj}(\mathcal{A})).$$

Hence $\operatorname{D}^{b}(\mathcal{A})$ is equipped with a weight structure with heart $\operatorname{Proj}(\mathcal{A}).$ This should be compared to the natural $t$-structure on $\operatorname{D}^{b}(\mathcal{A})$ with heart $\mathcal{A}.$
(3) A particularly interesting example of weight structures arises in the world of motives, namely for Voevodsky’s triangulated category of geometric motives $\operatorname{DM}_{gm}(k,\mathbb{Q})$ over a perfect field $k$, see [VSF00]. The existence of a $t$-structure on $\operatorname{DM}_{gm}(k,\mathbb{Q})$ is a notoriously difficult problem that implies, see [Bei10], for example Grothendieck’s standard conjectures.

Instead of a $t$-structure, Bondarko [Bon10] showed the existence of a weight structure $w$ called Chow weight structure on the category $\operatorname{DM}_{gm}(k,\mathbb{Q})$ whose heart

$$\operatorname{DM}_{gm}(k,\mathbb{Q})^{w=0}\cong\operatorname{Chow}(k,\mathbb{Q})$$

is equivalent to the category of Chow motives. The category of Chow motives was introduced by Grothendieck and has an elementary definition, see [Mil12]. Namely, one first considers the category of correspondences of smooth projective varieties up to rational equivalence. Here, objects are smooth projective varieties $X$ over $k$ and morphisms from $X$ to $Y$ are elements in the rational Chow group

$$\operatorname{CH}_{\dim(Y)}(X\times Y)_{\mathbb{Q}}.$$

Morphisms are composed via convolution. The category $\operatorname{Chow}(k,\mathbb{Q})$ is then obtained from the category of correspondences by idempotent completion and tensor-inverting the Lefschetz motive $\mathbb{L}=\ker(\mathbb{P}_{k}^{1}\to\operatorname{Spec}(k)).$ In other words, objects of weight zero in $\operatorname{DM}_{gm}(k,\mathbb{Q})$ arise from motives of smooth projective varieties.